beta;-normal topological spaces

Abstract

A topological space X is called π-normal if for any two disjoint closed subsets A and B of X one of which is π-closed, there exist two open disjoint subsets U and V of X such that A ⊆ U and B ⊆ V. We will present some characterizations of π-normality and some examples to show relations between π-normality and other weaker version of normality such as mild normality, almost normality, and quasi-normality. We investigate in this paper a weaker version of normality called π-normality. We will prove that π-normality is a property which lies between almost normality and normality. We will present some characterizations of π-normality and some examples to show relations between π-normality and other weaker versions of normality such as mild normality, almost normality, and quasi-normality. We will denote an ordered pair by x, y , the set of positive integers by N and the set of real numbers by R. A T 4 space is a T 1 normal space and a Tychonoff space is a T 1 completely regular space. The interior of a set A will be denoted by intA, and the closure of a set A will be denoted by A.

On βκ-normal spaces

“…A finite union of regular open sets is called π-open set and a finite intersection of regular closed sets is called π-closed set. The finite union (intersection) of π-closed sets is π-closed, but the infinite union (intersection) of π-closed sets need not be π-closed [12]. A point x ∈ X is called a δ-limit point of A if every regularly open neighborhood of x intersects A.…”

“…5. π-normal [12] if for any two disjoint closed subsets A and B of X one of which is π-closed, there exist two open disjoint subsets U and V of X such that A ⊂ U and B ⊂ V .…”

“…The notion of a normal topological space has been of interest to topologists for many years (see [2,7,8,12,24]) and plays an prominent role in general topology. Normality behaves differently from the other separation axioms in terms of subspaces and products.…”

“…Normality behaves differently from the other separation axioms in terms of subspaces and products. Several generalized notions of normal spaces studied in literature such as almost normal [21], π-normal [12], ∆-normal [7] (semi-nearly normal [18]), weakly ∆-normal [7], weakly functionally ∆-normal [7], quasi-normal [12], nearly normal [19] and densely normal [1], κ-normal [24] (or mildly normal [23]). Some of these forms are used to serve as a necessary ingredient towards a decomposition of normality (see [7,8,14]).…”